Transpiration cooling for a vehicle with low radius leading edge

ABSTRACT

A transpiration cooling system for avoiding overheating of an airfoil is provided. The airfoil is provided with a plurality of apertures and a source of pressurized fluid for providing a flow of fluid through the apertures to establish an aerodynamic radius. The aerodynamic radius of curvature of leading edge is sufficiently greater than the mechanical radius of curvature of the leading edge that peak heat flux is independent of the mechanical radius of curvature. The mechanical radius of curvature is preferably less than 50% of the aerodynamic radius of curvature during hypersonic operation. Preferably the mechanical radius of curvature is the smallest allowed by the fabrication method (i.e., a knife edge), such as being less than about 0.02 inches, preferably less than about 0.01 inches. The transpiration blowing rate can be adjusted so that the blowing rate and aerodynamic radius of curvature are relatively low except during periods of maximum heat flux, such as the shock-on-lip point. Since the mechanical radius of curvature is effectively zero, control of the aerodynamic radius of curvature provides complete control of peak heat flux. By adjusting blowing rate to the minimum necessary at any given velocity, transpiration consumption, drag and fuel injection are reduced or eliminated.

This is a divisional application of U.S. Ser. No. 07/956,928, filed Oct.5, 1992 now U.S. Pat. No. 5,351,917.

BACKGROUND OF THE INVENTION

The present invention relates to a method and apparatus fortranspiration cooling of a portion of the vehicle and in particular tocooling of a leading edge having a low radius of curvature.

High velocity vehicles such as air-breathing hypersonic vehiclestypically must be provided both with a system for minimizing drag andfor cooling portions of the vehicle, to accommodate heating rates.Accommodating both these requirements is particularly difficult forvehicles intended for sustained high velocity flight and/or orbitalinsertion, as opposed to reentry vehicles. Requirements during sustainedflight and/or orbital insertion include operation over extended periodsof time in an extremely hostile thermal environment. Stagnationtemperatures for vehicles approaching orbital velocities can exceed10,000° K. Attempts to achieve high net thrust (i.e., thrust minusdrag), lead to designs in which a vehicle may be required to operate ina high dynamic pressure (high thrust) environment with sharp leadingedges (to provide low drag). As discussed below, this combination ofhigh temperature, high pressure and small leading edge radius is,generally, considered antagonistic to attempts to cool the vehiclessince sharp leading edges are thought to produce extremely high localheat fluxes. In contrast, the thermal problem facing reentry vehicles isdifferent because in reentry vehicles, high drag is desirable or atleast acceptable. This allows a large leading edge radius to be used,accommodating installation of cooling systems. Furthermore, because ofthe typically short exposure of reentry vehicles to high temperatureenvironments, the use of transient cooling systems, such as a heat sinkor ablative techniques is feasible.

As noted above, in some respects the requirements of providing low dragand providing cooling are antagonistic. As the leading edge radius ofcurvature is reduced, the total heat load on the vehicle is reduced, aswell as the drag. Unfortunately, the peak heat flux (as opposed to thetotal heat load) increases as the radius of that portion decreases. Ingeneral, the peak heat flux on the leading edge (Q_(LE)) is inverselyproportional to the square root of the radius of curvature of theleading edge (R_(LE)), i.e., Q_(LE) ∝(R_(LE))^(-1/2).

One type of cooling system used for portions of vehicles is aregenerative system, in which a heat transfer fluid (typically the fuel)contacts the interior surface of a vehicle skin, absorbing heattherefrom, and flows to the engine. However, as the leading edge radiusof curvature is reduced, it becomes more difficult to install aregenerative circuit because of the small radius and slender sectionavailable for flow of the heat transfer fluid. Additionally, because theregenerative system depends on heat transfer through the skin, theregenerative or "back side convective" cooling system is limited by thethermal resistance of the structural material of the vehicle.

Another cooling system which has been used is transpiration cooling. Ina transpiration cooling system, there is no return flow of thetranspirant to the source. In transpiration cooling, fluid is conveyedto the interior surface of the vehicle skin and permitted to flowthrough perforations or pores through the vehicle skin. At a low rate offlow (or "blowing"), the transpirant fluid dilutes the hot boundarylayer, reducing the driving enthalpy and (normally) the heat flux. Thisreduction in heat flux is referred to as "partial blockage". At higherblowing rates, the hot boundary layer is pushed completely away from thesurface. Under these conditions, the surfaces are exposed only to thecoolant temperature and the heat flux (ignoring radiation) is reduced tozero (providing full blockage). However, operating a transpirationcooling system at a blowing rate sufficient to continuously provide fullblockage produces certain undesirable effects. Because the transpirantflow rate is high, there is a large consumption of transpirant and theweight of the large volume of transpirant which must be carriedincreases the size of the fuel tank needed for the vehicle. Further, thetranspiration system produces an amount of drag which increases as theblowing rate increases. Furthermore, when the leading edge being cooledis in the vicinity of the vehicle engine (e.g., the leading edge of theengine cowl) a high blowing rate leads to high fuel ingestion into theengine inlet.

The above described cooling problems are compounded in the case ofportions of the vehicle which may be subjected to shock interactions. Anexample is a hypersonic engine cowl. When the vehicle passes through itsinlet design mach number, the vehicle fore body shock(s) interact withthe cowl shock. This interaction from the two shocks (referred to as"shock-on-lip") produces a supersonic jet that sweeps across the cowlleading edge. Heat flux increases of 20 times have been reported toresult from impingement of this jet on the surface of the cowl.Accordingly, there is a need for a cooling system which can accommodateheat flux increases from a supersonic jet without providing anunacceptable amount of coolant consumption, drag, or fuel ingestion,over the duration of a flight or mission.

SUMMARY OF THE INVENTION

The present invention includes the recognition that the above-describedantagonism between increasing the cooling and decreasing the drag arisesbecause the peak heat flux is strongly influenced by radius of curvatureof the leading edge. The present invention involves an apparatus andmethod wherein the peak heat flux is made independent of the leadingedge radius of curvature. For these purposes, a distinction is madebetween the leading edge radius of curvature (referred to as the"mechanical radius of curvature" R_(m)), i.e., the radius of curvatureat the leading edge of the solid aircraft structure or skin and the"aerodynamic radius of curvature R_(a) ", i.e., the equivalentaerodynamic shock generating shape. It has been found that by making themechanical radius of curvature sufficiently small (compared to theaerodynamic radius) the peak heat flux becomes independent of themechanical radius of curvature and instead becomes a function almostentirely of the aerodynamic radius of curvature. In this situation, fromthe point of view of the freestream flow considering peak heat flux,there is no mechanical leading edge, (the mechanical radius of curvatureis effectively zero) i.e., the leading edge is equivalent to a leadingedge having a mechanical radius of curvature of zero. The peak heat fluxis substantially independent of the mechanical radius of curvature when:

R_(a) >R_(m)

The mechanical radius is effectively zero when it is less than 50% ofthe aerodynamic radius.

Once the mechanical radius of curvature is sufficiently small that it isuncoupled from peak heat flux, it can be designed to achieve the desiredreduction in drag without the penalty of a proportional increase in theblowing rate (and thus without the disadvantages of increased coolantconsumption, high drag and fuel ingestion). In one embodiment, themechanical radius of curvature is less than about 0.01 inches (about0.25 mm). To reduce drag, the mechanical radius of curvature of theleading edge is preferably the smallest allowed by the fabricationmethod (i.e., a knife edge).

Furthermore, because the aerodynamic radius is a function of the blowingrate, the aerodynamic radius can be adjusted during a flight. Inparticular, it is possible to maintain a relatively low blowing rateexcept during the portion of flight in which a supersonic jet impingeson the leading edge. The blowing rate can be increased to provide anaerodynamic radius that is sufficient to hold the supersonic jet off themechanical or structural surface of the vehicle. At higher or lowerspeeds, when the supersonic jet does not impinge, the blowing rate canbe reduced, thus causing an overall reduction in the amount of coolantconsumption, drag, and fuel ingestion, on a per-flight basis.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a view of a hypersonic vehicle;

FIG. 2 is a schematic view of shock interaction taken along line 2--2 ofFIG. 1;

FIG. 3 is a schematic, cross-sectional view of the leading edge belowthe shock-on-lip velocity;

FIG. 4 is an enlarged view of region 4--4 of FIG. 3;

FIG. 5 is a schematic depiction of a transporation cooling system,according to an embodiment of the present inventions;

FIG. 6 is a schematic cross-sectional view of a leading edge at a speedjust below the shock on limit condition;

FIG. 7 is an enlarged view of region 7--7 of FIG. 6;

FIG. 8 is a schematic cross-sectional view of a leading edge at avelocity about equal to the shock-on-lip point;

FIG. 9 is an enlarged view of region 9--9 of FIG. 8.

FIG. 10 is an enlarged view of region 10--10 of FIG. 9 showing the jetstand-off mechanism according to the present invention; and

FIG. 11 is a schematic view of the relationship between mechanicalradius, peak heat flux potential and aerodynamic radius according to anembodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

As seen in FIG. 1, a vehicle such as a hypersonic aircraft 10 has anumber of leading edges with respect to the ambient atmosphere asdefined by the freestream direction 12. Among the leading edges are theforward edges of the engine cowling 14. Depending upon the configurationand speed of the vehicle 10, certain of the leading edges may besubjected to shock interactions. As seen in FIG. 2, the engine cowl 14has a leading edge surface 16. The sharpness of the leading edge can bedefined by a radius of curvature of the leading edge 16. Since, on agiven scale of analysis, the contour of the leading edge may not beprecisely circular, references throughout to a radius of curvature arefor the purposes of relative measure of leading edge bluntness.

As shown in FIG. 2, in a region near the leading edge 16, severalaerodynamic shock waves can form and interact. In the situation depictedin FIG. 2, a cowl bow shock 18 interacts with an incident forebody shock20 to produce a region of supersonic jet flow 22 which impacts theengine cowl 14 in the region of the leading edge 16. Supersonic jets ofthis type are discussed in Allan R. Wieting, "Shock Interference Heatingin Scramjet Engines," AIAA Second International Aerospace PlanesConference, Oct. 29-31, 1990, incorporated by reference.

FIG. 3 depicts, schematically, in cross section, the cowl leading edgein which the aft portion 24 is regeneratively cooled and the forwardportion 26 is transpiration cooled as described more fully below. Thethickness 28 at the junction of the aft 24 and forward 26 sections, inthe embodiment depicted in FIG. 3, is about one inch (about 2.5centimeters). FIG. 3 depicts the situation in which the speed of thecowl 23 in the ambient fluid is less than the speed at which asupersonic jet impinges the leading edge (the "shock-on-lip" speed). Athypersonic speeds below the shock-on-lip point, a shock 30 is formed byinteraction of the leading edge 32 with the ambient fluid. Moreprecisely, the shock 30 is formed by the interaction of the aerodynamicradius 34 generated by the transpirant. The region of the leading edge32 is depicted in greater detail in FIG. 4. As shown in FIG. 4, theaerodynamic radius 34 is spaced away from the mechanical surface of thecowl 23 by transpiration cooling. A schematic view of the transpirationcooling system is depicted in FIG. 5. As shown in FIG. 5, thetranspirant flows to the surface of the cowl 36 through a large numberof pores 38a-f. These pores can be formed e.g., by a photochemicalmachining process. In this process thin sheets of metal are etched withthe coolant passages. These sheets or platelets can be diffusion bondedto form a solid structure with many small controlled passages. Theseflow circuits can include distribution manifolds 40 and individualmetering orifices 48. These orifices 48 control the flow to eachindividual pore to match the local coolant flow to the local heat fluxwhich can vary greatly across the surface. The heat flux and coolantflow requirements are generally greatest at the leading edge. At highblowing rates the coolant travels further from the surface before beingswept away. The interface 34 demarks the boundary between the coolantand free stream air. The metering of the coolant flow through theindividual pores has a strong influence on the shape of the boundary.

The transpirant is usually drawn from the main fuel 42 where it is usedto cool the engine or other aircraft hot surface 44. A regulating valve46 controls the amount of coolant delivered to the distribution manifold40. A number of fluids can be used as the transpirant, includingnitrogen gas, N₂ O₄, and water. Preferably, hydrogen is used. The rateof flow of transpirant through the holes 38a-38f is provided bycontrolling the opening of the valve 46, such as by a control mechanism50 in response to, for example, a speed indication from anemometer 52.

Returning to FIG. 4, at speeds below the shock-on-lip point, coolantflow can be minimal since there is no supersonic jet impaction and thusthe blowing rate need only be high enough to provide transpirationcooling sufficient to deal with the shock 30.

FIG. 6 shows the cowl 23 at a speed just below the shock-on-lipcondition. The heat flux in this condition is somewhat greater than theheat flux in the condition depicted in FIG. 4, and accordingly, coolantflow is increased (e.g., by opening valve 46 somewhat), pushing theaerodynamic radius 34 farther away from the structural surface inpreparation for the shock-on-lip condition. Thus, the aerodynamic radiusof FIGS. 3 and 4 is less than the aerodynamic radius of FIGS. 6 and 7.The mechanical radius of curvature of the leading edge is, of course,unchanged. The mechanical radius of curvature of the leading edge 32 issubstantially less than the aerodynamic radius of curvature 34.According to one embodiment of the present invention, the mechanicalradius of curvature of the leading edge 32 is sufficiently small thatthe freestream 12 is aerodynamically substantially unaffected by themechanical leading edge 32, instead interacting aerodynamically (atleast with regards to heat flux) as if only the aerodynamic radius 34were present. In one embodiment, the radius of curvature of the leadingedge 32 is sufficiently small that even though the aerodynamic radius 34may be one to two orders of magnitude larger than the mechanical radiusof curvature, the aerodynamic radius of curvature 34 is still relativelysmall such as being less than about 0.2 inches (about 5 millimeters),preferably less than about 0.1 inches (about 2.5 millimeters).

In one embodiment the aerodynamic radius of curvature is at least 1.5times said mechanical radius of curvature, preferably two times saidmechanical radius of curvature, more preferably at least five times saidmechanical radius of curvature, and most preferably at least about tentimes said mechanical radius of curvature. Although in the situationsdepicted in FIGS. 6 and 7 the flow of transpirant 56 is greater thanthat of the situation depicted in FIGS. 3 and 4, it is still relativelylow compared to that needed to cool a larger leading edge radius,contributing to minimizing the amount of transpirant 42 which must bestored on board.

FIG. 8 depicts the situation in which the cowl has a velocity equal tothe shock-on-lip point. In this situation, as depicted in FIG. 9, asupersonic jet 60 is formed as a result of the cowl shock 30 interactingwith the incident of forebody shock 62. The flow through the two obliqueshocks is compressed more efficiently and the result of the high energyis manifested in the high velocity of the flow confined to the jet 60.An interface region 64 forms as a result of the counter-flows oftranspirant 56 and the freestream 12. The jet 60 arrives at the coolantinterface 64 with the same strength as if it were approaching solidsurface of the same radius as the interface 64. However, because theinterface 64 is spaced from the mechanical surface of the cowl 23, thejet would have to penetrate further through the coolant layer 66 toreach the mechanical surface. As the jet penetrates into the coolantlayer 66, it continues to lose strength. The shock and viscous losses ofthe jet increase as the jet descends deeper into the higher-pressurecounter-flowing coolant 56. As the jet gets closer to the surface (theorigin of the coolant flow), the dynamic pressure of the coolant isgreater (the flow field surrounding the tip of the cowl is nearly aradially expanding source with increasing velocity and decreasingpressure as it moves away from the source). As depicted in FIG. 10, ifthe coolant flow is sufficient, equilibrium is established with the jet60 stagnated a "stand off" distance 68 away from the surface of the cowl23. This stand off distance 68 should be great enough that theindividual coolant jets emerging from each pore 38 have merged to form asolid front. If the stagnation is allowed to occur too close to thesurface, some of the hot gas from the jet 60 may find its way into"weak" wake zones (between the flow from individual coolant jets) andreach the surface. A safe stand off distance 68 will be determined bythe spacing between the coolant pores or slots and the pressure ratio.

Other dissipation mechanisms may further help in breaking up thesupersonic jet 60. Turbulence in the coolant jets may produce anunsteady bow shock 30. This unsteadiness may cause the jet 60 to movearound rapidly, causing stronger interactions between the jet 60 and thesurrounding flow field.

Further acceleration will cause the cowl to have a velocity greater thanthe shock-on-lip velocity. After passing through the shock-on-lipvelocity, the coolant flow can be throttled back to the value needed toreduce the heat flux without shock interaction (similar to the situationdepicted in FIGS. 6 and 7). Thus, mission coolant consumption may besignificantly reduced when compared to previous approaches.

Because previous designs have resulted in heat flux which is, at leastpartly, a function of mechanical radius of curvature, it was notpreviously possible to vary the radius of curvature to both reduce dragand provide effective cooling since the mechanical radius of curvatureof the leading edge could not be readily varied. According to thepresent invention, the mechanical radius of curvature is made extremelysmall so that (aerodynamically and considering heat flux) it isnon-existent (i.e., the imposed heat flux is an inverse function ofblowing rate). In this way, it is possible to provide effectivetranspiration cooling when it is most needed (at the shock-on-lipvelocity) without maintaining a blowing rate so high as to haveunacceptable transpirant consumption with the attendant increased dragand fuel ingestion.

FIG. 11 illustrates the manner in which the present invention providesfor a peak heat flux which is substantially independent of mechanicalradius. FIG. 11 is a schematic depiction of both mechanical radiusversus aerodynamic radius and mechanical radius versus the heat fluxpotential. FIG. 11 is schematic in nature and the axes are in arbitraryunits. Previous workers in the field recognize that high heat fluxproblems were made more serious in the case of the leading edge with asmall radius of curvature, as discussed above, principally because, asthe radius of curvature decreases, a given amount of heat isconcentrated in a smaller and smaller region. For this reason, previousworkers approaching the problem tended to work in design having arelatively high mechanical radius. It has been found, unexpectedly that,according to the present invention, many of the heat flux difficultiescan be solved by operating in a very low mechanical radius region 80.According to one embodiment of the invention the mechanical radius isless than 0.05 inches (about 1.25 mm), preferably less than about 0.01inches (about 0.25 millimeters). Operation in this region is practicalbecause, for very low mechanical radius of curvature, the heat flux isno longer a function of mechanical radius but solely a function ofaerodynamic radius. This is an advantage because, unlike the mechanicalradius of curvature, the aerodynamic radius of curvature can be varied.The independence of heat flux from mechanical radius in this region canbe seen from FIG. 11. In FIG. 11, a first curve 70 shows heat fluxpotential as a function of mechanical radius when transpiration-cooled.Curve 72 shows aerodynamic radius as a function of mechanical radius fora transpiration-cooled surface with a sufficient blowing rate to coolthe surface. The dotted curve 74 shows values for which the aerodynamicradius is equal to the mechanical radius (no blowing). As seen in FIG.11, in regions of larger mechanical radius 78, the heat flux 70 issubstantially an inverse function of mechanical radius. However, in thelow mechanical radius region 80, the heat flux curve 70 is substantiallyconstant and independent of the mechanical radius curve 74. Thus, in thelow mechanical radius region 80, changes in mechanical radius havesubstantially no effect on the peak heat flux potential 70.

In view of the above discussion, a number of advantages of the presentinvention may be seen. The present invention permits transpirationcooling of a hypersonic vehicle without excessive transpirantconsumption, drag or fuel ingestion. Thus, large cooling rates arerestricted to a limited period (during which the cowl passes through theshock-on-lip point) so that overall transpirant consumption for a givenmission or flight can be relatively small. The present invention permitsa small mechanical radius which will result in reduced drag whileproviding sufficient transpiration cooling to deal with the shock-on-lipproblem. A number of variations and modifications of the invention canalso be used. The present invention can be used in connection withsurfaces other than an engine cowl leading edge, fuel injection struts,such as wing and fuselage surfaces. Certain aspects of the invention canbe used without employing other aspects. For example, it is possible toprovide for low mechanical radius of curvature devices withtranspiration cooling where coolant flow is not necessarily reduced atvelocities above the shock-on-lip velocity. The present invention can beused in connection with applications where a supersonic shock other thana shock-on-lip jet is formed. The present invention can be used inconnection with vehicles when no supersonic jet is formed.

Although the present invention has been described by way of a preferredembodiment and certain variations and modifications, other variationsand modifications can also be used the invention being defined by thefollowing claims.

What is claimed:
 1. In a hypersonic aircraft having an engine and anengine cowl, the engine cowl having a skin, a leading edge of saidengine cowl having a mechanical radius of curvature, a transpirationcooling system for avoiding overheating of said cowl, comprising:aplurality of pores in said skin permitting flow of fluid through saidportion of said skin; means for providing pressurized fluid to aninterior surface of said skin for flow through said skin, forming anaerodynamic radius of curvature; said aerodynamic radius of curvaturebeing greater than said mechanical radius of curvature and saidmechanical radius being about 0.01 inches or less.
 2. A cooling system,as claimed in claim 1, wherein said aerodynamic radius of curvature isabout 0.1 inches.
 3. A transpiration cooling system for an airfoil, atleast a leading edge part of said airfoil defining a first mechanicalradius of curvature, the cooling system comprising:first means forallowing fluid flow through at least a portion of the surface of saidairfoil; second means for supplying a flow of fluid to said first means;third means for controlling the rate of flow of fluid through saidportion of the surface to define an aerodynamic radius of curvature,said aerodynamic radius of curvature being at least 1.5 times largerthan said mechanical radius of curvature so that peak heat flux isdetermined entirely by said aerodynamic radius of curvature and issubstantially unaffected by said mechanical radius of curvature.
 4. Atranspiration cooling system for an airfoil, at least a leading edgepart of said airfoil defining a first mechanical radius of curvature,the cooling system comprising:first means for allowing fluid flowthrough at least a portion of the surface of said airfoil; second meansfor supplying a flow of fluid to said first means; third means forcontrolling the rate of flow of fluid through said portion of thesurface to define an aerodynamic radius of curvature, said aerodynamicradius of curvature being larger than said mechanical radius ofcurvature so that drag is determined substantially entirely by saidaerodynamic radius of curvature and is substantially unaffected by saidmechanical radius of curvature; wherein said mechanical radius ofcurvature is about 0.01 inches or less.
 5. A cooling system, as claimedin claim 4, wherein said means for controlling includes means forcontrolling the rate of flow to provide an increased rate of flow duringimpingement of a supersonic jet on said airfoil.
 6. A cooling system, asclaimed in claim 4, wherein, when said airfoil is moving at a velocitywith respect to the ambient atmosphere, said means for controllingincludes means for providing a first flow rate at a first airfoilvelocity, a second flow rate at a second airfoil velocity greater thansaid first airfoil velocity and a third flow rate at a third airfoilvelocity greater than said second airfoil velocity, said second flowrate being greater than said first flow rate and said third flow ratebeing less than said second flow rate.
 7. In an airfoil having avelocity with respect to ambient atmosphere, said airfoil having amechanical surface, said airfoil being impinged by a supersonic jet whenit reaches a second airfoil velocity after accelerating through a firstvelocity, said first velocity lower than said second velocity, a methodfor cooling said airfoil comprising:accelerating said vehicle to saidsecond velocity, said airfoil surface being subjected, during saidacceleration, to a supersonic jet at said second velocity of saidvehicle, flowing a fluid from the interior of said airfoil to theexterior to create an aerodynamic surface; and controlling the rate offlow to provide a higher rate of flow at said second velocity than atsaid first velocity; wherein the rate of flow at said second velocityprevents said jet from impinging on said mechanical surface.
 8. Amethod, as claimed in claim 7, wherein the rate of flow at said secondvelocity is greater than the rate of flow at said first velocity, saidsecond velocity being greater than said first velocity.
 9. A method, asclaimed in claim 7, wherein:said step of flowing comprises flowing saidfluid through a plurality of openings to define a merged surface atwhich the individual flows from a plurality of said openings merge. 10.A method, as claimed in claim 9, wherein the rate of flow at said secondvelocity prevents said jet from penetrating said merged surface.
 11. Ina hypersonic vehicle having an air breathing engine and an engine cowl,the engine cowl having a skin, the leading edge of said engine cowlhaving an airfoil surface, a method for providing for orbitalacceleration of said vehicle while avoiding overheating of the enginecowl, the method comprising:accelerating said vehicle from a firstvelocity to an earth-orbital velocity, said airfoil surface beingsubjected, during said acceleration, to a supersonic jet when saidvehicle attains a second velocity, flowing a fluid from the interior ofsaid airfoil to the exterior during at least a part of said accelerationto create an aerodynamic surface, wherein said fluid flows through aplurality of apertures formed in said skin; and controlling the rate offlow to provide a higher rate of flow at said earth-orbital velocitythan at said first velocity.